Two postings: News about a possible proof of Poincare Conjecture,
and Mimura and Toda on the naming of the elements in the homotopy
groups of spheres............DMD
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Subject: proof of Poincare/Geometrization Conjecture?
Date: Mon, 18 Nov 2002 18:26:05 -0500
From: Zbigniew Fiedorowicz
My colleague, Dan Burghelea, believes that there is a major development,
if not a complete proof of Thurston's Geometrization Conjecture for
3-manifolds (and hence also of the Poincare Conjecture). The paper in
question is by Grisha Perelman (of the Steklov Institute in St. Petersburg),
entitled "The entropy formula for the Ricci flow and its geometric applications",
which was deposited in the Math. Archive on Nov. 11 and is available via the
link http://front.math.ucdavis.edu/math.DG/0211159.
The paper does not make any explicit claims about proving the above conjectures,
but does claim to prove the conjectures in section 6 of Richard Hamilton's paper
"Formation of singularities in the Ricci flow" in Surveys in Diff. Geom. 2 (1995),
(which unfortunately is not available in our library). Moreover Perelman refers to
another 1999 paper of Hamilton on the latter's program to prove the Geometrization
Conjecture and concludes his introduction with the following tantalizing statement:
"We have not been able to confirm Hamilton's hope ... ; still we are able to show
that ...; by our earlier (partly unpublished) work this is enough for topological
conclusions."
Perelman is a well respected differential geometrer who is regarded as an
expert on Ricci flow.
I wonder if anyone has comments or further information.
Actually I overlooked the following explicit statement in Perelman's
paper: "Finally in Section 13 we give a brief sketch of the proof of
the geometrization conjecture."
Zig Fiedorowicz
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